The East Lansing Model: a Bayesian uncertainty quantified optical potential for rare isotopes

The East Lansing Model presents a global, Bayesian uncertainty-quantified optical potential for neutrons and protons that utilizes a novel asymmetry-dependent parameterization calibrated with (n,n), (p,p), and (p,n) experimental data to significantly reduce extrapolation uncertainties toward the proton and neutron driplines.

The Gist

The Big Picture: Predicting the Unpredictable

Imagine you are a chef trying to cook a meal for a crowd of people you've never met before. You have a recipe book that works perfectly for your regular customers (stable, common elements like Carbon or Oxygen). But now, you need to cook for a group of exotic, rare guests (rare isotopes found in exploding stars or nuclear reactors) that you've never seen in a kitchen before.

If you just guess based on your old recipes, you might burn the food or serve something inedible. In science, this is called extrapolation. When we try to predict how rare, unstable atoms behave based on data from stable ones, our guesses often come with huge "what if?" uncertainties.

The East Lansing Model (ELM) is a new, super-smart recipe book designed specifically to handle these rare guests. It doesn't just guess; it calculates the odds of being right, giving scientists a much clearer picture of how the universe works in extreme environments.

The Problem: The "One-Size-Fits-All" Trap

For decades, physicists have used "Optical Potentials" to predict how particles (like neutrons or protons) bounce off atomic nuclei. Think of an atomic nucleus as a billiard ball. When another ball hits it, it scatters in a certain direction.

• The Old Way: Scientists used models that assumed the "shape" of the force between a neutron and a proton was exactly the same as the force between a proton and a proton. They treated them like identical twins.
• The Flaw: In reality, these particles aren't identical twins; they are more like cousins. They act similarly, but they have subtle differences, especially when you get to the "edge of the map" (the limits of stability).
• The Missing Ingredient: Previous models mostly ignored a specific type of experiment called a charge-exchange reaction (where a proton turns into a neutron during a collision). It's like trying to understand how a car drives on ice only by watching it drive on dry pavement. You miss the crucial data about how it slips.

The Solution: The East Lansing Model (ELM)

The team at Michigan State University built a new model with three major upgrades:

1. Giving the "Twins" Their Own Wardrobes

In the old models, the "isoscalar" (the part of the force that treats neutrons and protons the same) and the "isovector" (the part that treats them differently) were forced to wear the same clothes (the same mathematical shape).

The ELM Analogy: Imagine you are tailoring a suit. The old model forced the left arm and the right arm to be cut from the exact same piece of fabric, assuming they are identical. The East Lansing Model says, "Wait, the left arm might need to be slightly longer or wider." It gives the two parts of the force independent shapes. This allows the model to capture the subtle "neutron skin" (a layer of extra neutrons on the outside of heavy atoms) that older models missed.

2. Using the "Secret Sauce" Data

The team didn't just look at the standard data (how particles bounce off). They included the "charge-exchange" data (where particles swap identities).

The Analogy: Imagine you are trying to learn a new language. The old models only studied the grammar books (elastic scattering). The East Lansing Model also studied the slang and idioms (charge-exchange reactions). By including this "slang," the model understands the language much better, especially when speaking to native speakers in rare dialects (unstable isotopes).

3. The "Uncertainty GPS"

This is the most important part. The model uses Bayesian Uncertainty Quantification.

The Analogy:

• Old Models: Give you a single answer. "The particle will hit here." If you are wrong, you have no idea how wrong you are.
• East Lansing Model: Gives you a map with a "confidence zone." It says, "The particle will likely hit here, but there is a 95% chance it's within this circle." If the circle is small, you are very confident. If the circle is huge, the model admits, "I'm not sure, we need more data."

This is crucial for science. It tells researchers, "You can trust this prediction for nuclear energy," or "Don't trust this prediction for a supernova simulation; the uncertainty is too high."

Why Does This Matter?

1. The Origin of Heavy Elements: About half the heavy elements in the universe (like gold and uranium) are made in violent cosmic events like exploding stars or colliding neutron stars. These events involve rare, unstable atoms. To simulate these events on a computer, we need accurate recipes for how these atoms interact. ELM provides those recipes with much higher precision.
2. Nuclear Energy & Safety: When nuclear fuel breaks apart (fission), it creates neutron-rich fragments. Understanding how these fragments interact with neutrons is vital for reactor safety and waste management. ELM helps predict these interactions without needing to build a physical reactor for every single scenario.
3. Better Science, Less Guesswork: By admitting what they don't know (the uncertainty), scientists can stop wasting time on theories that are likely wrong and focus on the experiments that actually matter.

The Bottom Line

The East Lansing Model is like upgrading from a paper map to a real-time GPS with traffic alerts. It doesn't just tell you where the atoms are going; it tells you how sure it is about the route. By treating neutrons and protons as unique individuals rather than identical twins, and by listening to the "slang" of charge-exchange reactions, it allows us to explore the furthest, most unstable corners of the atomic world with confidence.

In short: It's a smarter, more honest way to predict how the building blocks of the universe behave when they are pushed to their absolute limits.

Technical Summary

1. Problem Statement

The primary challenge addressed is the extrapolation of nuclear reaction models to regions far from stability (near the proton and neutron driplines).

• Context: Astrophysical processes (e.g., supernovae, neutron star mergers) and fission applications rely on nuclear networks involving neutron-rich isotopes that have never been measured in a laboratory.
• Limitation of Current Models: Existing global optical potentials (e.g., KD, CH, KDUQ, CHUQ) are typically fitted to stable nuclei using elastic scattering data. When extrapolated to rare isotopes, these models suffer from:
  • Unknown Uncertainties: Data-driven methods (like neural networks) often produce unrealistic uncertainty estimates when extrapolating.
  • Inadequate Constraints: Traditional global potentials often ignore charge-exchange reactions (specifically $(p,n)$) or assume that the isoscalar and isovector components of the optical potential share the same geometric form factors. This assumption limits the model's ability to accurately describe neutron skins and asymmetry-dependent effects.
  • Poor Performance: Previous uncertainty-quantified models (KDUQ, CHUQ) show extremely wide credible intervals and poor agreement with $(p,n)$ data when extrapolating to neutron-rich systems.

2. Methodology

The authors developed the East Lansing Model (ELM), a global, spherical optical potential for nucleon scattering ($A \ge 40$, $E = 10-100$ MeV) using a rigorous Bayesian calibration framework.

A. Theoretical Framework (The Model)

• Lane Form: The potential $U(r)$ is constructed using the Lane form to preserve isospin symmetry:
  $$U(r) = U_0(r) + \frac{\tau \cdot T}{A} U_1(r)$$
  where $U_0$ is the isoscalar term and $U_1$ is the isovector term.
• Novelty in Geometry: Unlike previous models (e.g., CH) that assume $U_0$ and $U_1$ share the same radial form factors (radius and diffuseness), ELM allows them to have independent geometric parameters.
  • The real central potential is defined as: $V(r) = V_0 f_0(r) \pm \frac{N-Z}{A} V_1 f_1(r)$.
  • $f_0$ and $f_1$ are independent Woods-Saxon form factors with distinct depths ($V_0, V_1$), radii ($R_0, R_1$), and diffuseness ($a_0, a_1$).
• Parameter Count: The full ELM has 21 free parameters. A simplified version, ELM0, imposes identical geometry for isoscalar and isovector terms (18 parameters) to test the necessity of the independent geometry.

B. Data Curation

• Sources: Experimental data was assembled from EXFOR using open-source tools (exfor-tools).
• Dataset:
  • Elastic Scattering: 57 entries, 5,965 data points ($(n,n)$ and $(p,p)$).
  • Charge Exchange: 17 entries, 710 data points ($(p,n)$ to Isobaric Analog States, IAS).
  • Constraints: Spherical targets ($0^+$ ground states), beam energies $10 < E < 100$ MeV.

C. Statistical Calibration

• Bayesian Approach: The calibration solves for the posterior distribution of model parameters ($\alpha$) and model discrepancy hyperparameters ($\beta$).
  $$y_{obs} + \epsilon \equiv \zeta_{truth} = y_{model}(\alpha) + \delta_{discrepancy}$$
• Model Discrepancy: A parametric, diagonal model discrepancy is used, where the standard deviation is a fraction ($\beta_l$) of the average data/model output. This accounts for systematic errors not captured by the physics model.
• Sampling: Markov Chain Monte Carlo (MCMC) was used via the open-source library rxmc. The Schrödinger equation was solved using the jitR package (R-matrix method).
• Priors: Based on the CHUQ parameters, with specific priors for the new isovector geometry parameters derived from recent analyses.

3. Key Contributions

1. Independent Geometric Form Factors: The introduction of independent radius and diffuseness parameters for the isovector term ($U_1$) allows the model to encode neutron skin effects physically, rather than forcing them through artificial strength adjustments.
2. Integration of $(p,n)$ Data: The study demonstrates that $(p,n)$ charge-exchange data is essential for constraining the isovector geometry. Models calibrated without $(p,n)$ data (ELM0el) fail to converge or describe charge-exchange observables.
3. Unified Bayesian Framework: The development of a reproducible, open-source workflow (rxmc, jitR, exfor-tools) that simultaneously calibrates elastic and charge-exchange data while quantifying uncertainties and model discrepancy.
4. Improved Extrapolation: The ELM framework provides significantly reduced uncertainties when extrapolating to the limits of stability compared to KDUQ and CHUQ.

4. Results

• Parameter Constraints:
  • The inclusion of $(p,n)$ data forces the maximum likelihood to shift, significantly increasing the depth of the isovector term.
  • However, this shift is only physically consistent when the independent geometry is also allowed. The full ELM (purple in Fig. 1) shows distinct posterior distributions for isovector radii ($r_1$) and diffuseness ($a_1$) compared to isoscalar counterparts.
  • The isovector geometry is found to be slightly more diffuse than the isoscalar, with a radial intercept difference of $\sim 1 - 1.5$ fm.
• Predictive Performance:
  • Elastic Scattering: All models (ELM, ELM0, ELM0el) describe elastic scattering equally well.
  • Charge Exchange $(p,n)$: ELM0el (no $(p,n)$ data) fails to describe $(p,n)$ angular distributions. ELM0 (same geometry) describes them but requires unphysical parameter shifts. ELM (independent geometry + $(p,n)$ data) provides the best description with the most physical parameters.
  • Uncertainty Reduction: ELM achieves a $\sim 10\%$ reduction in fractional uncertainty for $(p,n)$ predictions compared to ELM0.
• Extrapolation to Rare Isotopes:
  • Neutron Total Cross Sections: For Sn isotopes ($A=100$ to $170$), ELM predictions differ significantly from KDUQ and CHUQ. ELM shows tighter credible intervals and better agreement with available data for neutron-rich isotopes (e.g., $^{132}$Sn).
  • Proton Reaction Cross Sections: Show less dependence on the isovector geometry changes, but ELM remains consistent.
  • Neutron Skins: The model predicts neutron skin thickness scaling as $R_{skin} \approx 1.59$ fm for $A \sim 48$ decreasing to $1.35$ fm for $A \sim 208$.

5. Significance

• Astrophysics & Nuclear Physics: The East Lansing Model provides a critical tool for nuclear astrophysics, enabling more reliable simulations of nucleosynthesis in explosive environments (supernovae, mergers) where neutron-rich nuclei dominate.
• Uncertainty Quantification: By rigorously quantifying uncertainties and demonstrating that they remain small even near the driplines, ELM addresses a major gap in current nuclear theory. It moves beyond "black box" extrapolations to physics-based, uncertainty-aware predictions.
• Methodological Advancement: The paper establishes that charge-exchange data is non-negotiable for constraining isovector potentials and that independent geometric form factors are necessary for a physically consistent description of neutron-rich matter.
• Open Science: The authors have released the model parameters, definitions, and the entire calibration workflow as open-source tools, facilitating future updates and community adoption.

Conclusion: The East Lansing Model represents a paradigm shift in optical potential modeling, moving from phenomenological fits on stable nuclei to a Bayesian, uncertainty-quantified framework that leverages charge-exchange data and independent isovector geometries to robustly predict nuclear reactions for rare isotopes.